Question

Write each expression without parentheses. Assume all variables are positive. $$ (2 p)^{5} $$

Short answer

Question: Simplify the expression (2p)^5 without using parentheses. Answer: 32p^5

Step-by-step solution

Identify the base and exponent

In the expression \((2p)^{5}\), the base is \(2p\) (inside the parentheses), and the exponent is \(5\).

Apply the power of a product property

Now we will apply the power of a product property: \((ab)^n = a^nb^n\). In our case, \(a = 2\), \(b = p\), and \(n = 5\). So we have \((2p)^{5} = 2^5p^5\).

Calculate the powers

Calculate the powers of the individual terms. We have \(2^5 = 32\) and \(p^5\) remains as it is.

Write the final simplified expression

Combine the results from Step 3 to write the simplified expression. Therefore, \((2p)^{5}\) can be written without parentheses as \(32p^5\).

Key concepts

Power of a Product Property

The power of a product property is essential when working with expressions that include exponents. When you have a term like \((ab)^n\), the power of a product property allows you to distribute the exponent to each factor inside the parentheses. This means that \((ab)^n = a^n \cdot b^n\). For example, in the expression \((2p)^5\), both 2 and \(p\) are raised to the power of 5.
This simplifies to \(2^5 \cdot p^5\). This rule is helpful because it breaks down complicated expressions into smaller, more manageable parts.
By understanding this property, you can simplify expressions more easily.
Remember, this property can only be used when you have a product inside the parentheses and a single exponent outside.
Be sure to apply the exponent to every factor within the parentheses.

Simplifying Expressions

Simplifying expressions is a crucial skill in algebra. It helps you reduce complex expressions into simpler ones, making them easier to evaluate or compare.
In the exercise, once you've applied the power of a product property, you get \(2^5 \cdot p^5\).
Next, you'll need to calculate the numerical power. Here, raising 2 to the 5th power gives you 32.

• Step 1: Identify each part of the expression.
• Step 2: Use properties like the power of a product to break it down.
• Step 3: Calculate powers and combine similar terms.

The simplified expression \(32p^5\) is much easier to work with because it removes parentheses and clearly shows the role of each factor. Simplifying helps in solving equations, comparing sizes, and more.

Exponentiation

Exponentiation is a mathematical operation involving numbers known as bases and exponents.
An exponent tells you how many times to multiply the base by itself.
In the problem \((2p)^5\), the exponent is 5, indicating that both the number 2 and the variable \(p\) need to be multiplied by themselves five times. Calculating \(2^5\) involves multiplying 2 by itself 4 more times:

• \(2 \times 2 = 4\)
• \(4 \times 2 = 8\)
• \(8 \times 2 = 16\)
• \(16 \times 2 = 32\)

Therefore, \(2^5 = 32\). Exponentiation is a powerful math tool that helps express large numbers compactly.
It's used extensively in scientific notation, compound interest calculations, and more.
Understanding exponentiation makes handling significant data and equations more efficient.